Properties

Label 12-1040e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.265\times 10^{18}$
Sign $1$
Analytic cond. $327991.$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s − 12·11-s + 4·19-s − 8·25-s + 4·29-s − 24·31-s + 4·41-s + 8·49-s + 4·59-s + 20·61-s + 16·71-s + 56·79-s + 4·89-s − 48·99-s − 28·101-s − 48·109-s + 18·121-s + 4·125-s + ⋯
L(s)  = 1  + 4/3·9-s − 3.61·11-s + 0.917·19-s − 8/5·25-s + 0.742·29-s − 4.31·31-s + 0.624·41-s + 8/7·49-s + 0.520·59-s + 2.56·61-s + 1.89·71-s + 6.30·79-s + 0.423·89-s − 4.82·99-s − 2.78·101-s − 4.59·109-s + 1.63·121-s + 0.357·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 5^{6} \cdot 13^{6}\)
Sign: $1$
Analytic conductor: \(327991.\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 5^{6} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3576357013\)
\(L(\frac12)\) \(\approx\) \(0.3576357013\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 8 T^{2} - 4 T^{3} + 8 p T^{4} + p^{3} T^{6} \)
13 \( ( 1 + T^{2} )^{3} \)
good3 \( 1 - 4 T^{2} + 16 T^{4} - 62 T^{6} + 16 p^{2} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 - 8 T^{2} + 88 T^{4} - 734 T^{6} + 88 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 + 2 T + p T^{2} )^{6} \)
17 \( 1 + 752 T^{4} - 50 T^{6} + 752 p^{2} T^{8} + p^{6} T^{12} \)
19 \( ( 1 - 2 T + 13 T^{2} - 116 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 70 T^{2} + 2127 T^{4} - 47860 T^{6} + 2127 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 2 T + 43 T^{2} - 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 12 T + 113 T^{2} + 664 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 160 T^{2} + 11392 T^{4} - 506230 T^{6} + 11392 p^{2} T^{8} - 160 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 2 T + 43 T^{2} + 156 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 84 T^{2} + 5136 T^{4} - 202462 T^{6} + 5136 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 216 T^{2} + 21176 T^{4} - 1243838 T^{6} + 21176 p^{2} T^{8} - 216 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 250 T^{2} + 28167 T^{4} - 1878700 T^{6} + 28167 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 2 T + 133 T^{2} - 276 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 10 T + 135 T^{2} - 1252 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 298 T^{2} + 41783 T^{4} - 3518604 T^{6} + 41783 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 8 T + 178 T^{2} - 936 T^{3} + 178 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \)
79 \( ( 1 - 28 T + 453 T^{2} - 4744 T^{3} + 453 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 322 T^{2} + 51623 T^{4} - 5250876 T^{6} + 51623 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 2 T + 223 T^{2} - 396 T^{3} + 223 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 10 T^{2} + 4687 T^{4} - 11980 T^{6} + 4687 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.22929923345128984807833066945, −5.20522751726269058906142269913, −5.07146360514980638407902723544, −4.88616550005978573055503193868, −4.62155925922653084813835949092, −4.56624544503193350585443123827, −4.14540115837022679818137044596, −3.93006140113071297937528891578, −3.88518222455757989429400743931, −3.73478896882496805718307156978, −3.66328283855190575728936141182, −3.60207337571398861250924555823, −3.25348830748725152905750582970, −2.82146726646765842954002779382, −2.75753180868428059750653058040, −2.62235500311961847470936753809, −2.33365597353771839960555146138, −2.32821882092073093696268526172, −2.06391933569241690183813928135, −1.77402623228512728587966297602, −1.68629399086473033117675737799, −1.24456946330371577187409920163, −0.912362153380320916157288364721, −0.61231164307133943569652619925, −0.11028309682381126372571330953, 0.11028309682381126372571330953, 0.61231164307133943569652619925, 0.912362153380320916157288364721, 1.24456946330371577187409920163, 1.68629399086473033117675737799, 1.77402623228512728587966297602, 2.06391933569241690183813928135, 2.32821882092073093696268526172, 2.33365597353771839960555146138, 2.62235500311961847470936753809, 2.75753180868428059750653058040, 2.82146726646765842954002779382, 3.25348830748725152905750582970, 3.60207337571398861250924555823, 3.66328283855190575728936141182, 3.73478896882496805718307156978, 3.88518222455757989429400743931, 3.93006140113071297937528891578, 4.14540115837022679818137044596, 4.56624544503193350585443123827, 4.62155925922653084813835949092, 4.88616550005978573055503193868, 5.07146360514980638407902723544, 5.20522751726269058906142269913, 5.22929923345128984807833066945

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.